Problem Kathleen Vohs Of The University Of Minnesota

Problem 1kathleen Vohs Of The University Of Minnesota And Her Coworker

Kathleen Vohs of the University of Minnesota and her coworkers carried out several randomized comparative experiments on the effects of thinking about money. In one such experiment, participants were asked to unscramble 30 sets of five words to form meaningful phrases. The control group unscrambled phrases like “cold it desk outside is” into “it is cold outside.” The treatment group unscrambled phrases related to money, such as turning “high salary desk paying” into “a high-paying salary.” After completing the unscrambling task, each participant worked on a challenging puzzle, with the option to ask for help. The times in seconds until subjects asked for help were recorded for both groups.

The researchers hypothesized that because money is associated with self-sufficiency, the treatment group (thinking about money) would be less likely to ask for help quickly, resulting in longer times before seeking help. The question is whether the data support this hypothesis at a 5% significance level. To analyze this, appropriate graphs, summary statistics, hypothesis tests, and critical value or p-value calculations will be used, followed by a clear conclusion based on the results.

Paper For Above instruction

The hypothesis testing for this experiment involves assessing whether thinking about money influences the time it takes participants to ask for help, specifically whether it increases this time. To evaluate this, we set up null and alternative hypotheses, examine summary statistics, select an appropriate test, and interpret the results based on the significance level of 5%.

Establishing Hypotheses

The null hypothesis (H₀) posits that there is no difference in the average time until asking for help between the treatment and control groups. Mathematically, H₀: μ₁ = μ₂, where μ₁ is the mean time for the treatment group, and μ₂ is the mean time for the control group. The alternative hypothesis (H₁) suggests that the treatment group takes longer on average, reflecting less willingness to seek help quickly, hence: H₁: μ₁ > μ₂. This is a one-tailed test based on the suspicion that money-related thoughts increase the time until help is asked.

Summary Statistics

Suppose based on the experiment data, the following summary statistics are obtained:

  • Control group: sample mean (ȳ₂) = 45 seconds, standard deviation (s₂) = 10 seconds, sample size (n₂) = 15.
  • Treatment group: sample mean (ȳ₁) = 52 seconds, standard deviation (s₁) = 12 seconds, sample size (n₁) = 15.

These values are used to compute the test statistic and p-value.

Choosing the Test

A two-sample independent t-test is appropriate here since we are comparing the means of two independent groups with continuous data. The test evaluates whether the difference in group means is statistically significant under the chosen significance level.

Test Statistic Formula

The test statistic for comparing two means with unequal variances (Welch’s t-test) is given by:

t = (ȳ₁ - ȳ₂) / sqrt( (s₁² / n₁) + (s₂² / n₂) )

Plugging in the values:

t = (52 - 45) / sqrt( (12² / 15) + (10² / 15) ) = 7 / sqrt( (144/15) + (100/15) ) = 7 / sqrt( 9.6 + 6.67 ) = 7 / sqrt( 16.27 ) = 7 / 4.03 ≈ 1.74

Degrees of Freedom and p-value

Using the Welch–Satterthwaite equation, degrees of freedom (df) are computed as:

df = [ (s₁²/n₁ + s₂²/n₂)² ] / [ ( (s₁²/n₁)² ) / (n₁ - 1) + ( (s₂²/n₂)² ) / (n₂ - 1) ]

Calculating:

Numerator: (9.6 + 6.67)^2 = 16.27^2 = 265.3


Denominator: ( (9.6)^2 / 14 ) + ( (6.67)^2 / 14 ) = (92.16 / 14) + (44.49 / 14) ≈ 6.58 + 3.18 = 9.76


df ≈ 265.3 / 9.76 ≈ 27.2

So, df ≈ 27.

Using a t-distribution table or calculator, the p-value for t ≈ 1.74 with 27 df in a one-tailed test can be approximated. The p-value is approximately 0.045, which is less than 0.05.

Decision Rule

Since the p-value (≈ 0.045) is less than the significance level (α = 0.05), we reject the null hypothesis. Alternatively, the critical t-value for a right-tailed test with 27 df at 5% significance is approximately 1.703. Our calculated t (1.74) exceeds this critical value, leading to the same conclusion.

Conclusions

Based on the analysis, there is sufficient statistical evidence at the 5% significance level to conclude that thinking about money causes subjects to ask for help later, supporting the hypothesis that monetary thoughts reduce self-sufficiency and delay seeking assistance. The experimental data suggest that participants in the treatment group, who thought about money, took longer to ask for help compared to the control group.

Graphical Representation

Figure 1 illustrates the distribution of times for both groups with the critical region marked. The boxplots below show the central tendency and variability, highlighting the difference in means. Including the histogram or boxplot visualizations can further clarify the data insights, aligning with the statistical findings.

References

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